Baptism for the dead. Also known as “baptism by proxy” or “vicarious baptism,” this Ordinance is performed in Mormon temples by living church members on behalf of someone who is deceased. The soul for whom the work is done is given a chance to receive the gospel in Spirit Prison, though acceptance by the deceased person is not guaranteed. According to Joseph Smith, “The greatest responsibility in this world that God has placed upon us is to seek after our dead” (Teachings of the Prophet Joseph Smith, p. 356). Tenth President Joseph Fielding Smith explained, “Since the requirement of obedience to the Gospel ordinances is made of all men, and since they cannot enter into the kingdom without complying with the law the Lord has given, a work must be done in behalf of those who have died without knowledge of the Gospel and its requirements, and who never had the opportunity of repentance and remission of sins” (The Way to Perfection, p. 152). And fifteenth President Gordon B. Hinckley stated, “Through living proxies who stand in behalf of the dead, the same ordinances are available to those who have passed from mortality. In the spirit world they then are free to accept or reject those earthly ordinances performed for them, including baptism, marriage, and the sealing of family relationships. There must be no compulsion in the work of the Lord, but there must be opportunity” (Be Thou an Example, p. 131). A number of deceased famous people have had baptisms performed in Mormon temples. See here.
Although Paul mentions a practice known as baptism for the dead in 1 Corinthians 15:29, little is really known on the subject other than he appears to exclude himself from such a practice by using the word they. The Mormon interpretation is rejected as are all ideas of a post-mortem salvation. Hebrews 9:27 clearly shows “it is appointed unto men once to die, but after this the judgment.” This makes an after-death restitution with God an impossibility. Ironically, this same idea is taught in the Book of Mormon in Alma 34:34, 35.